In Exercises 54–67, solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. sin 2x = √ 3 sin x

Verified step by step guidance

1. Start by rewriting the given equation: \(\sin 2x = \sqrt{3} \sin x\).
2. Use the double-angle identity for sine: \(\sin 2x = 2 \sin x \cos x\). Substitute this into the equation to get \(2 \sin x \cos x = \sqrt{3} \sin x\).
3. Bring all terms to one side: \(2 \sin x \cos x - \sqrt{3} \sin x = 0\). Factor out \(\sin x\): \(\sin x (2 \cos x - \sqrt{3}) = 0\).
4. Set each factor equal to zero and solve separately: 1) \(\sin x = 0\) 2) \(2 \cos x - \sqrt{3} = 0\).
5. For \(\sin x = 0\), find all \(x\) in \([0, 2\pi)\) where sine is zero. For \(2 \cos x - \sqrt{3} = 0\), solve for \(\cos x = \frac{\sqrt{3}}{2}\) and find all \(x\) in \([0, 2\pi)\) that satisfy this.

Key Concepts

Double-Angle Identity for Sine

The double-angle identity expresses sin(2x) as 2 sin(x) cos(x). This allows rewriting the equation sin 2x = √3 sin x into a form involving sin(x) and cos(x), facilitating algebraic manipulation and solution finding.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within the given interval. It often requires factoring, using identities, and considering the periodic nature of sine and cosine functions.

Interval Restriction and Exact Values

Solutions must be found within the interval [0, 2π), meaning all valid angles between 0 and 2π are considered. Using exact values (like π/3, π/6) is preferred, but approximate decimal values to four decimal places are acceptable when exact forms are complex.